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Unformatted text preview: Chapter 7: Proof Systems: Soundness and Completeness Introduction Proof systems are built to prove statements. Proof systems are an inference machine with special statements, called provable state ments being its final products. The starting points of the inference are called axioms of the system . We distinguish two kinds of axioms: logi cal AL and specific SP . 1 Semantical link : we usually build a proof system for a given language and its seman tics i.e. for a logic defined semantically. First step : we choose as a set of logical ax ioms AL some subset of tautologies, i.e. statements always true. A proof system with only logical axioms AL is called a logic proof system . Building a proof system for which there is no known semantics we think about the logi cal axioms as statements universally true. 2 We choose as axioms a finite set the state ments we for sure want to be universally true, and whatever semantics follows they must be tautologies with respect to it. Logical axioms are hence not only tautolo gies under an established semantics, but they also guide us how to establish a se mantics, when it is yet unknown. The specific axioms SP are these formulas of the language that describe our knowl edge of a universe we want to prove facts about. Specific axioms are not universally true, they are true only in the universe we are inter ested to describe and investigate. 3 A proof system with logical axioms AL and specific axioms SP is called a formal the ory . The inference machine is defined by a finite set of rules, called inference rules . The inference rules describe the way we are allowed to transform the information within the system with axioms as a staring point. The transformation process is called a for mal proof and can be depicted as follows: AXIOMS RULES applied to AXIOMS Provable formulas RULES applied to any expressions above NEW Provable formulas . ..... . etc. ..... . The provable formulas are those for which we have a formal proof are called con sequences of the axioms. 4 Semantical link : the rules have to preserve the truthfulness of what we are proving. Rules with this property are called sound rules and the system a sound proof system . Soundness Theorem : for any formula A of the language of the system S , If a formula A is provable in a logic proof system S , then A is a tautology. Formal theory with specific axioms SP , based on a logic defined by the axioms AL is a proof system S with logical axioms AL and specific axioms SP . Notation : TH S ( SP ). 5 Soundness Theorem for formal theory says: for any formula A of the language of the theory TH S ( SP ), If a formula A is provable in the theory TH S ( SP ) , then A is true in any model of the set of specific axioms SP ....
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This note was uploaded on 02/12/2011 for the course CSE 541 taught by Professor Bachmair,l during the Spring '08 term at SUNY Stony Brook.
 Spring '08
 Bachmair,L
 Computer Science

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